Exploration of stochastic dynamics and complexity in an epidemic system.

We investigate the effect of noise in an epidemic system. We have studied dynamics and complexity for both the deterministic and its noise-induced model. We have verified the stochastic sensitivity under the variation of noise strength and changing the initial conditions of the noise-induced system. It confirms that noise can make significant perturbation in the stochastic sensitivity. To quantify the dynamics, phase space analysis is done under both noisy and noise free conditions. The transition between regular and chaotic dynamics has been examined by 0 - - 1 test. Corresponding complexity analysis is also done using the weighted recurrence entropy method. Numerical results confirm the chaotic dynamics in the noise-induced epidemic system within a larger region of parameters compared to the same in its noise free part.

Introduction

Ecosystem is one of the highly nonlinear systems that can be observed in many real word phenomena [1]. A nonlinear system can produces various kind of long-term dynamics [2-7]. Several studies have been done for the ecosystems, that show existence of regular as well as chaotic long-term dynamics [8–15, 27, 28]. The most well-known chaotic illustration can be given by the logistic map-a discrete system of animal reproduction [8]. Chaotic behaviour has been also observed for the continuous ecological system in many literature [9–19, 26]. In [27], it has been proved that chaos is one of the destabilizing factors that can collapse ecosystems. Further, a stabilizing effect of chaos is also observed in [28] that implies re-immigration or rescue effect in the ecosystem. Thus, chaotic dynamics is an obvious property in ecosystems and hence the corresponding future is almost unpredictable [17–19, 26].

Chaos in a nonlinear system is such a state that shows sensitivity with initial condition [3-5]. In that case, the system losses its memory with a small perturbation of the initial condition [3-5]. Chaos can be investigated for both deterministic and stochastic system [3–7, 29–33]. For a deterministic system, Lyapunov analysis is one of the efficient methods for finding chaos in the corresponding system [3-5]. To investigate chaos in a stochastic system, the test method is one of the effective nonlinear tools [34-37].

A chaotic system always has unpredictable behaviour. So uncertainty is a ubiquitous property of this system. For a stochastic phenomenon, it is more than a deterministic phenomenon. It means that the uncertainty increases as the system becomes more and more random. The measure of uncertainty was first introduced by C.E. Shannon [38]. Several entropy measures have been developed and applied widely in diverse domains of research [20-25]. Complexity in a system can be characterized by measuring disorder in the corresponding phase space [39-45]. Different types of measures have been proposed to quantify dynamical complexity of a system [41-48]. A weighted recurrence based measure has been proposed in [48], which can describe the different structural patterns of the phase space. The proposed weighted entropy measure correlates to the dynamics of the system (discrete or continuous) [48].

Noise is an inherent feature that appears in different forms in the real world phenomena. Several research have been established on the effect of noise in different systems [29–33, 45]. Further, the effect of noise in the epidemic has been already established in the literature [16-18]. So, noise-induced long-term analysis is one of the best possible studies to approximate the dynamics of a stochastic epidemic system. In [26], the chaotic phenomenon and its relation with the diseases have been investigated in the context of generalist predators. The effect of noise is not examined for the system given in [26]. We, therefore, studied the impact of noise on an epidemic system using measures of chaos and complexity.

This article is organized as follows: In section 2, a stochastic epidemic system is constructed. Section 3 described stochastic sensitivity [53] of the noise induced epidemic system. Using test method, dynamical fluctuation is analyzed over the feasible ranges of disease reproduction and noise strength. The whole discussion is given in Sect. 4. Section 5 discusses the complexity of the phase spaces based on weighted recurrence entropy analysis. Finally, a conclusion is given in Sect. 6.

Construction of stochastic epidemic model

We consider an epidemic model, proposed in [26], given bywhere are healthy and infected variables of the susceptible species respectively. The variable represents population of the immune prey species. The last variable indicates predator population of the system. The parameters a, b represent the reproduction rate of the prey species. R denotes the basic reproduction number of the disease. The parameters c and d is defined in [26] by ( are rates of individual prey species) and (, are the predator starvation rate in the absence of prey and death rate of infected individual prey species respectively). A complete pictorial representation of the system (1) is given in Fig.  1.

Fig. 1

Representing the conversion of deterministic (SYSTEM-1) to noise-induced (SYSTEM-2) epidemic system. Green arrow indicates the conversion from SYSTEM-1 to SYSTEM-2. Left arrow with sign indicates the induction of noise to the healthy population rate . The corresponding noise affected growth is given by . The SYSTEM-1 and SYSTEM-2 are given by (1) and (2) respectively. The curved arrows associated with the species are indicated intermediate relations

To construct the stochastic version of (1), we incorporate noise for the healthy population rate. The noise is taken as noise. Incorporation of noise in the deterministic system and its converted system is shown in Fig. 1. The reason for inducing the -noise can be stated as follows: (i) -noise has long-range dependence [49-52]. It affects the long-term dynamics of the system, ii) fluctuation of -noise can be observe in many physical phenomena [49-52]. As the dynamics of an epidemic depend on the interaction of prey and predator populations, the effect of -noise is an obvious phenomenon in the epidemic system. The constructed noise-induced epidemic model is then given bywhere function represents power noise and K its strength respectively. We fix as taken in [26]. We only variate R, K by and respectively.

In the following section, we investigate stochastic sensitivity of the system (2) under the variation of noise strength (K) and initial condition.

Stochastic sensitivity

To quantify stochastic sensitivity, we consider 4D attractors of the system (2) with the initial condition . As this system possesses an equilibrium state for [26], we consider (1, 0.2, 0.4, 0.1) as an equilibrium point and call it . According to the definition of quasipotential [53], the function v(x) of the 4D attractors is given bywhere . It can be approximated by Gaussian density function of the random trajectory flow in the attractors, wherewhere and N being a normalized constant [53].

The density function of the random state in the neighbourhood of can be geometrically described by a confidence ellipsoid [53]. The confidence ellipsoid is defined aswhere P represents fiducial probability. Here, is an inverse of which is defined byFor , is given by and the corresponding 4D confidence ellipsoid is defined aswhere ( being eigenvector for the eigenvalues of the stochastic sensitivity matrix W). As s measure stochastic sensitivity of a system, is also effective for the same. We call as and calculate its fluctuation over () and (, ). The corresponding oscillations are shown in Fig. 2a and b respectively. From Fig. 2a, it can be observed that has nonlinear increasing trend for (). It indicates sensitivity of (2) increases nonlinearly with the increasing noise strength. On the other hand, a sharp increasing trend in can be observed in Fig.2b for with , . It implies increasing stochastic sensitivity in (2) for a small perturbation of . Further, oscillations in are calculated over . The corresponding contour plot is visualized in Fig. 2c. From the figure, it can be seen that the is increasing according to increase of noise strength K within a small neighourhood of . So, this analysis indicates effectiveness of power noise on increasing the stochastic sensitivity in the system (2).

Fig. 2

a and b represents vs. K and plots respectively. Variation in with are given in c. The associate colour bar indicates values of

We now investigate the dynamics of the system (2) under the variation of basic reproduction number (R) of the disease and noise strength (K).

Quantifying dynamics of the noise free and noisy epidemic model

To quantify the dynamics, we first characterize the asymptotic dynamics of (2) using phase space analysis. The phase space analysis is done under the variation of both and respectively. Some of the phase spaces are projected in 2D (see the Fig. 3). From Fig. 3a, periodic orbit can be observed. It indicates regular asymptotic dynamics of the system at . On the other hand, irratic movement in phase trajectory can be observed in 3b-d. It indicates complex asymptotic dynamics of the same system with the respective cases . In this way, we have examined nature of long-term dynamics of (2) for and . However, chaos cannot be confirmed from these phase space analysis. To characterized regular and chaotic dynamics, we have implemented chaos test method.

Fig. 3

(a)–(d) represent 2D phase space of the system (2) with . Corresponding (p, q) plots are given in (e)-(f) respectively. To calculate p and q, we have considered solution component of (2)

In this method, a solution (L being the length of the solution) is considered of a system. Then, is decomposed into two components , whereIn (4), the value of can be chosen from the interval for each n . The cloud consists of can be characterized by two types of geometric-regular and Brownian motion like structures [34-37]. The regular and Brownian motion like structures indicates non-chaotic and chaotic signature in the system respectively [34-37]. The cloud of is known as (p, q)-plot.

We investigate (p, q)-plots for the system (2) for and . Some of the plots are shown in Fig.  3e-h at respectively.

From the figures, it can be observed that the deterministic structure can only be found in Fig. 3e. The remaining (p, q)-plots shows always Brownian motion like geometry. So, Fig. 3e indicates non-chaotic dynamics at and Fig. 3f-h indicate chaos in (2) with respectively. Further, chaos is quantified for both the system (1)and (2) with (fixed K) and (fixed R) respectively.

To quantify chaos, we have implemented a measure (Based on ) proposed by Gottwald et.al [35], which can quantifies chaotic as well as non-chaotic dynamics of a system, whereHere, is called mean square displacement of the components , given byFigure 4a shows fluctuation of over with (blue) and (red) respectively. It can be observed from the figure that, and over the respective and . However, for all can also be observed in the same figure. It indicates chaotic paradigm of the system (2) increases over compare to the same of (1). On the other hand, Fig. 4b indicates that the system can have stochastic chaos for noise strength with the fixed . From both the observations, it can therefore assure that the noise can enhance complexity in the system (1). Further, oscillation in over is investigated. We only show the oscillation for . The corresponding contour plot is given in Fig. 4c. From the figure, it can be investigated that system (1) can produced chaos for most of the region . It can also verified that the same system shows chaotic phenomena for the region . So, the entire analysis confirms existence of larger chaotic paradigm can be obtained for the system (2) compare to same in (1).

Fig. 4

a Represents vs. R() curves with (blue) and (red) respectively. b Represents vs. K oscillation with . -matrix plot is given in c with . The coloured bar indicates values of

To investigate complexity of the system, we have applied recurrence weight based entropy method [48]. The following section discusses on finding complexity in (1) with noise free and noisy conditions.

Quantifying complexity of the noise free and noisy epidemic model

Weighted recurrence (WR) is first developed by Eroglu et al, based on finite dimensional phase space of a nonlinear system [48]. For an n-dimensional phase space (N being the length of the trajectory of the phase space), WR is denoted by and defined asHere, represents distance between two points and it is calculated by . As quantifies dispersion between the trajectories, disorder in the phase space can be characterized by -matrix plot. However, values of are always lies between 0 and 1 as it is defined as exponential decay of . We observed nature of -matrix plots with the variation of and . Figure 5a-d represents some of the -matrix plots of the system (2) with . From Fig. 5a, it can be observed that there exists only two variation in the -matrix. It indicates almost homogenous trajectory’s pattern in the corresponding phase space. On the other hand, various patterns for the can be investigated in Fig. 5b–d. It signifies heterogeneous movements in the respective phase trajectory. In this way, we have characterized disorder in the system dynamics for and . Similarly, probability distribution is further investigated by constructing sample space (T being number of events). For , the corresponding vs. curves are shown in Fig. 5e-f respectively. From the figures, it can be seen that only two peaks are exist in Fig. 5e. It indicates only two variation in and hence no disorder in the corresponding phase space. However, large number of peaks can be observed in all the remaining figures (see Fig. 5f-h). It signifies highly disordered phase spaces for the system (2) with . So, results of probability distribution correlates with the nature of respective -matrix plots as well as system dynamics.

Fig. 5

a Weighted recurrence plots for the system (2) with respectively. The corresponding probability density curves are shown in b. To count probability , we have considered 50 bins

Hence, disorder in the system (2) can be characterized using weighted recurrence.

To quantify the disorder, we used weight recurrence entropy (S) given bywhere calculated between ( being the small perturbation in ).

Figure 6a shows fluctuation in S over with (blue) and (red) respectively. It can be observed from the figure that, is increasing with (). It indicates that the complexity of the non chaotic dynamics is lies between 0.5 and 2.8. It can be also observed that with (). Further, oscillation in S is always greater than 3 can be seen in the same figure. On the other hand, Fig. 6b indicates that the system produces complexity greater than 3 for with the fixed . Both the phenomena, strongly correlates with the dynamical fluctuations shown in Fig. 5a and b. In the next, we compute oscillation in S over . Fig. 6c shows fluctuation in S only for . From the figure, it can be investigated that the system (1) reveals phase spaces with for most of the region . The region of can also be verified over the region . The entire analysis is thus strongly correlated with the dynamical variation shown in Fig. 4c.

Fig. 6

a Represents S vs. R() curves with (blue) and (red) respectively. b Represents S vs. K oscillation with . S(K, R)-matrix plot is given in c with . The coloured bar indicates values of S(K, R)

Conclusion

In this research, our main aim is to find the effect of noise in a nonlinear epidemic system and its complexity. To assure the impact of noise on the system, a stochastic sensitivity analysis is performed. The results indicate that noise strength is one of the major factors to enhance chaos in the system. Biologically it means a stable epidemic system can be more unpredictable at some extension of noise. The dynamical fluctuation was analyzed under the variation of K and R. It confirms chaotic dynamics can be observed in the system for a wide range of (K, R). Finally, weighted entropy analysis is applied to investigate the complexity of both the noise free and noisy epidemic systems. It also assures a larger region of high complexity for the noise induced system than the same for the noise free system. As complex dynamics correspond to a destabilizing ecosystem, our analysis confirms that the noisy epidemic system can be more unpredictable compared to the same in its deterministic part. In this way, we have established the effect of noise on the epidemic system.